Answer
$E = \frac{\rho~R^2}{2~\epsilon_0~r}$
Work Step by Step
We can draw a Gaussian cylinder of length $L$ with the axis along the same axis as the charged cylinder.
We can find the electric field at a distance $r$ from the cylinder axis where $r \gt R$:
$\epsilon_0~\Phi = q_{enc}$
$(\epsilon_0)~(E)~(2\pi~r~L) = \rho~\pi~R^2~L$
$(\epsilon_0)~(E)~(2~r) = \rho~R^2$
$E = \frac{\rho~R^2}{2~\epsilon_0~r}$